I have seen two ways of describing the ring structure of modular (with respect to the full $SL (2, \mathbb Z)$) forms: $\mathbb C [E_4, E_6]$ and $\mathbb C [E_4,E_6, \Delta]/(E_4^3-E_6^2-1728\Delta )$, where $E_4$ and $E_6$ are the normalized Eisenstein series of weight 4 and 6, respectively.
As far as I've always known, every level 1 modular form is a polynomial in $E_4$ and $E_6$ so the first description is completely appropriate. The second one has the virtue of promoting $\Delta$ to a special place in the ring, but since $E_4$, $E_6$, and $\Delta$ do satisfy the well known relationship $E_4^3-E_6^2=1728\Delta$ nothing is really gained. So, unless I'm missing something, these two presentations are equivalent.
My question is then about a distinction that seems to be made in a paper by Sati: OP2 Bundles in M-theory. In section 3 he describes the ring of modular forms as $\mathbb Z [E_4,E_6, \Delta]/(E_4^3-E_6^2-1728\Delta )$ and he states, by results of Mahowald and Hopkins, that there are certain manifolds which map under the Witten genus to the ring $\mathbb Q [E_4, E_6]$, indicating that these are modular forms "without the discriminant."
I guess I don't really understand what this last part means. Can somebody please clarify this issue for me? Thanks in advance.